Let G be a non-compact, real semisimple Lie group. We consider maximal complexifications of G which are adapted to a distinguished one-parameter family of naturally reductive, left-invariant metrics. In the case of G=SL2(ℝ) their realization as equivariant Riemann domains over Gℂ=SL2(ℂ) is carried out and their complex-geometric properties are investigated. One obtains new examples of non-univalent, non-Stein, maximal adapted complexifications.